Loop Group Decompositions in Almost Split Real Forms and Applications to Soliton Theory and Geometry

TL;DR

This paper establishes a global Birkhoff decomposition for almost split real forms of loop groups with compact underlying Lie groups, enabling new applications in soliton theory, geometry, and harmonic maps.

Contribution

It introduces a global decomposition result for loop groups in almost split real forms, with significant implications for integrable systems and harmonic map representations.

Findings

Global Birkhoff decomposition for almost split real forms proved

Dressing actions are global for compact cases in integrable systems

Infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps established

Abstract

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the $U$-hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1 wave maps) from surfaces into compact symmetric spaces. An "Iwasawa-type" decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given.